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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 
"" {TEXT 256 7 "RECORDS" }}{PARA 259 "" 0 "" {TEXT 259 11 "Tevian Dray
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 257 34 "co
pyright 1998-2000 by Tevian Dray" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 260 "" 0 "" {TEXT 258 36 "What does the phonograph record see?" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 
14 "Initialization" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restar
t:with(linalg):with(plots):with(plottools):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 2594 "# Record\n\n# Release 4:\n#Machine:=substring(in
terface(version),39..41):\n# Release 5.1:\n#Machine:=substring(interfa
ce(version),41..43):\n# Fixed?\nMachine:=IBM:\n\nArrow:=proc(base,dir,
color)\n  local Dir:\n  global Machine:\n  if Machine=`IBM` or Machine
=`DEC` or Machine=`SUN` then\n    Dir:=base+dir\n  else\n    Dir:=dir
\n  fi;\n  arrow(base,Dir,0.02,0.05,0.1,color)\nend:\n\nrsol2:=vector(
[cos(Omega*t)*x[0]+cos(Omega*t)*t*v[x]\n\011+sin(Omega*t)*y[0]+sin(Ome
ga*t)*t*v[y]-rho,\n\011-sin(Omega*t)*x[0]-sin(Omega*t)*t*v[x]+\n\011co
s(Omega*t)*y[0]+cos(Omega*t)*t*v[y]]):\n\ndd:=disk([0,0],1,color=wheat
):\nrho:=0:\n\nFix:=proc(pos,vel,vframe,time,steps,rot)\n  local G,tt,
aa,MM,rsol,ll,ff,av,i,vx,vy,rsol3,t:\n  if evalb(vframe=`ROT`) then\n \+
   av:=evalm(vel-rot*[pos[2],-pos[1]]):\n    vx:=av[1]:\n    vy:=av[2]
:\n  else\n    vx:=evalm(vel)[1]:\n    vy:=evalm(vel)[2]:\n  fi:\n  rs
ol3:=[cos(rot*t)*pos[1]+cos(rot*t)*t*vx\n\011+sin(rot*t)*pos[2]+sin(ro
t*t)*t*vy,\n\011-sin(rot*t)*pos[1]-sin(rot*t)*t*vx\n\011+cos(rot*t)*po
s[2]+cos(rot*t)*t*vy]:\n  G:=[seq(0,i=0..steps)]:\n  G[1]:=display(Arr
ow([0,0],[-0.25,0],color=green),dd):\n  for i from 1 to steps do\n    \+
tt[i]:=rot*i*time/steps:\n    aa[i]:=Arrow(pos,[vx,vy]*i*time/steps,co
lor=blue):\n    MM[i]:=matrix([[cos(tt[i]),-sin(tt[i])],[sin(tt[i]),co
s(tt[i])]]):\n    rsol[i]:=multiply(MM[i],rsol3):\n    ll[i]:=plot([rs
ol[i][1],rsol[i][2],t=0..i*time/steps],\n\011\011color=red,thickness=2
):\n    ff[i]:=Arrow([0,0],[-0.25*cos(tt[i]),-0.25*sin(tt[i])],color=g
reen):\n    G[i+1]:=display(ll[i],aa[i],ff[i],dd):\n  od:\n  display(G
,scaling=constrained,insequence=true,axes=NONE);\nend:\n\nRot:=proc(po
s,vel,vframe,time,steps,rot)\n  local H,f2,t2,M2,l2,Start2,v2,a2,av,i,
vx,vy,rsol3,t:\n  if evalb(vframe=`ROT`) then\n    av:=evalm(vel-rot*[
pos[2],-pos[1]]):\n    vx:=av[1]:\n    vy:=av[2]:\n  else\n    vx:=vel
[1]:\n    vy:=vel[2]:\n  fi:\n  rsol3:=[cos(rot*t)*pos[1]+cos(rot*t)*t
*vx\n\011+sin(rot*t)*pos[2]+sin(rot*t)*t*vy,\n\011-sin(rot*t)*pos[1]-s
in(rot*t)*t*vx\n\011+cos(rot*t)*pos[2]+cos(rot*t)*t*vy]:\n  H:=[seq(0,
i=0..steps)]:\n  H[1]:=display(Arrow([0,0],[-0.25,0],color=green),dd):
\n  f2:=Arrow([0,0],0.25*[-1,0],color=green):\n  for i from 1 to steps
 do\n    t2[i]:=rot*i*time/steps:\n    M2[i]:=matrix([[cos(t2[i]),sin(
t2[i])],[-sin(t2[i]),cos(t2[i])]]):\n    l2[i]:=plot([rsol3[1],rsol3[2
],t=0..i*time/steps],color=red,thickness=2):\n    Start2[i]:=evalm(mul
tiply(M2[i],pos)):\n    v2[i]:=evalm(multiply(M2[i],[vx,vy]*i*time/ste
ps+pos)-Start2[i]):\n    a2[i]:=Arrow([Start2[i][1],Start2[i][2]],[v2[
i][1],v2[i][2]],color=blue):\n    H[i+1]:=display(l2[i],a2[i],f2,dd):
\n  od:\n  display(H,scaling=constrained,insequence=true,axes=NONE);\n
end:\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 15 "Getting started" }}{PARA 0 "" 0 "" {TEXT -1 216 "Straight
 line motion in the plane traces out a complicated path as seen by a r
otating observer.  The needle of a phonograph moves radially towards t
he center of the record, yet traces out a long spiral on the record!" 
}}{PARA 0 "" 0 "" {TEXT -1 223 "Replace the record by a blank sheet of
 paper, and the phonograph needle by a red pen.  Furthermore, allow th
e needle to start anywhere, and move in any fixed direction with const
ant speed.  What curve is drawn on the paper?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We showed in class that t
he equation of this curve is" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
44 "r[rel]:=rsol2[1]*r[hat]+rsol2[2]*theta[hat];" }}}{PARA 0 "" 0 "" 
{TEXT -1 40 "where the initial position of the pen is" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 18 "r[0]:=[x[0],y[0]];" }}}{PARA 0 "" 0 "" 
{TEXT -1 27 "and its initial velocity is" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "v[fix]:=[v[x],v[y]];" }}}{PARA 0 "" 0 "" {TEXT -1 35 
"(A derivation of this equation for " }{XPPEDIT 18 0 "r[rel]" "6#&%\"r
G6#%$relG" }{TEXT -1 80 ", as well as the differential equation it sol
ves, can be found in the worksheet " }{HYPERLNK 17 "2D Rotations" 1 ":
usersB:tevian:maple:ph429:2Drot.mws" "" }{TEXT -1 49 "; the equation i
s also derived in Class Notes 1.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "v[x]" "6#&%\"
vG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v[y]" "6#&%\"vG6#%\"yG
" }{TEXT -1 49 " denote the initial velocity with respect to the " }
{TEXT 260 11 "nonrotating" }{TEXT -1 37 " frame.  The initial velocity
 in the " }{TEXT 261 8 "rotating" }{TEXT -1 18 " frame is given by" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "v[rel]:=eval(subs(t=0,diff(r
[rel],t)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 "Demos" }}
{PARA 0 "" 0 "" {TEXT -1 151 "The following demos graph the path of th
e pen, both as seen by the nonrotating observer (blue vector), and as \+
seen by the rotating observer (red line)." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "First, we set define some paramete
rs.  The angular velocity of the record (in radians/second " }{TEXT 
263 16 "counterclockwise" }{TEXT -1 14 ") is given by " }{XPPEDIT 18 
0 "Omega" "6#%&OmegaG" }{TEXT -1 21 "; a typical value is " }{XPPEDIT 
18 0 "Omega=1" "6#/%&OmegaG\"\"\"" }{TEXT -1 46 ".  The number of seco
nds the demo will run is " }{XPPEDIT 18 0 "T" "6#%\"TG" }{TEXT -1 21 "
, a typical value is " }{XPPEDIT 18 0 "T=5" "6#/%\"TG\"\"&" }{TEXT -1 
44 ".  The total number of pictures is given by " }{XPPEDIT 18 0 "N" "
6#%\"NG" }{TEXT -1 39 ", so that the time between pictures is " }
{XPPEDIT 18 0 "T" "6#%\"TG" }{TEXT -1 1 "/" }{XPPEDIT 18 0 "N" "6#%\"N
G" }{TEXT -1 27 ".  A reasonable choice for " }{XPPEDIT 18 0 "N" "6#%
\"NG" }{TEXT -1 4 "  is" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "N:
=5;" }}}{PARA 0 "" 0 "" {TEXT -1 55 "(A smoother animation is obtained
 for larger values of " }{XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT -1 2 "; " 
}{XPPEDIT 18 0 "N=10" "6#/%\"NG\"#5" }{TEXT -1 41 " works fairly well.
  But large values of " }{XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT -1 74 " ma
y also take significantly longer to run.  See what works well for you.
)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "Fina
lly, you should make a choice of how you want the graphics to be displ
ayed, using any " }{TEXT 262 3 "one" }{TEXT -1 25 " of the following 3
 lines" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plotsetup(inline):
" }{TEXT -1 26 "     (graphs in worksheet)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "plotsetup(window):" }{TEXT -1 29 "     (graphs in M
aple window)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "plotsetup(x
11):" }{TEXT -1 41 "            (graphs in separate X window)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 378 "In the figures you will constr
uct, the blue arrow represents the linear motion seen by an inertial o
bserver, while the red curve represents the motion seen by the rotatin
g observer.  (The green arrow is to help you keep track of the rotatio
n.)  Think of the tip of the blue arrow as containing a red pen which \+
writes on the record as the record rotates beneath the (moving) pen." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 166 "The de
mo consists of 2 functions, one from the point of view of each referen
ce frame.  The first of these shows what the nonrotating observer sees
, and takes the form" }}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "Fix(start,velocity,vframe,T,N,Omega" "6#-%$FixG6(%&startG%)velocity
G%'vframeG%\"TG%\"NG%&OmegaG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 6 "Here, " }{XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT -1 2 ", " }{XPPEDIT 
18 0 "T" "6#%\"TG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Omega" "6#%&OmegaG
" }{TEXT -1 23 " are as defined above, " }{XPPEDIT 18 0 "start" "6#%&s
tartG" }{TEXT -1 31 " denotes the initial position, " }{XPPEDIT 18 0 "
velocity" "6#%)velocityG" }{TEXT -1 35 " denotes the initial velocity,
 and " }{XPPEDIT 18 0 "vframe" "6#%'vframeG" }{TEXT -1 79 " specifies \+
whether the initial velocity is with respect to the fixed observer (" 
}{MPLTEXT 0 21 13 "vframe:=`FIX`" }{TEXT -1 28 ") or the rotating obse
rver (" }{MPLTEXT 0 21 13 "vframe:=`ROT`" }{TEXT -1 2 ")." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "For example, try" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fix([-1,0],[1/4,0],`FIX`,4
,N,-10);" }}}{PARA 0 "" 0 "" {TEXT -1 55 "which is roughly what happen
s when you play a record.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
263 "" 0 "" {TEXT -1 242 "(To run the animations, use the buttons at t
he top; for inline display, you will first need to click on the figure
.  For either inline or Maple window display, you can find out what th
e buttons do by turning on Balloon Help in the Help menu.)" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "To see this from \+
the reference frame of the rotating observer, use the command" }}
{PARA 261 "" 0 "" {XPPEDIT 18 0 "Rot(start,velocity,vframe,T,N,Omega" 
"6#-%$RotG6(%&startG%)velocityG%'vframeG%\"TG%\"NG%&OmegaG" }{TEXT -1 
1 " " }}{PARA 0 "" 0 "" {TEXT -1 67 "which takes the same parameters. \+
 Compare your previous output with" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "Rot([-1,0],[1/4,0],`FIX`,4,N,-10);" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "You can reproduce the r
esults in Figure 10-4 on page 389 of Marion and Thornton using " }
{XPPEDIT 18 0 "Omega=1" "6#/%&OmegaG\"\"\"" }{TEXT -1 25 ", the initia
l  positition" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "MTstart:=[-
1/2,0]:" }}}{PARA 0 "" 0 "" {TEXT -1 46 "one of the initial velocities
 (and total time " }{XPPEDIT 18 0 "T" "6#%\"TG" }{TEXT -1 15 " as indi
cated):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "MTv[a]:=[0,1]*3/2
:" }{TEXT -1 31 "                       (T=0.86)" }{MPLTEXT 1 0 19 "\n
MTv[b]:=[0,1]*4/5:" }{TEXT -1 30 "                       (T=2.9)" }
{MPLTEXT 1 0 20 "\nMTv[c]:=[0,1]*9/20:" }{TEXT -1 29 "                \+
     (T=17.3)" }{MPLTEXT 1 0 21 "\nMTv[d]:=[0,1]*0.328:" }{TEXT -1 26 
"                   (T=5.0)" }{MPLTEXT 1 0 28 "\nMTv[e]:=[1,1]*0.47/sq
rt(2):" }{TEXT -1 12 "    (T=3.83)" }{MPLTEXT 1 0 29 "\nMTv[f]:=[1,1]*
0.283/sqrt(2):" }{TEXT -1 9 "  (T=3.3)" }}}{PARA 0 "" 0 "" {TEXT -1 
15 "and specifying " }{MPLTEXT 0 21 5 "`ROT`" }{TEXT -1 5 " for " }
{MPLTEXT 0 21 6 "vframe" }{TEXT -1 30 ".  Try it!  Try other choices!
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Fix(MTstart,MTv[e],`ROT`
,3.83,N,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}}{MARK "2" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }